INVARIANT TWINS it must have been ripe port, to judge by what Sylvester got out of the decanter. Cayley and Sylvester came together again professionally when Cayley accepted an invitation to lecture at Johns Hopkins for half a year in 1881-82. He chose Abelian functions, in which he was researching at the time, as his topic, and the sixty-seven- year-old Sylvester faithfully attended every lecture of Ms famous friend. Sylvester had still several prolific years ahead of Mm, Cayley not quite so many. We shall now briefly describe three of Cayley's outstanding contributions to mathematics in addition to his work on the theory of algebraic invariants. It has already been mentioned that he invented the theory of matrices, the geometry of space of n dimensions, and that one of his ideas in geometry threw a new light (hi Klein's hands) on non-Euclidean geometry. We shall begin with the last because it is the hardest. Desargues, Pascal, Poncelet, and others had created projec- tive geometry (see chapters 5,13) in which the object is to dis- cover those properties of figures which are invariant under projection. Measurements - sizes of angles, lengths of lines - and theorems which depend upon measurement, as for example the Pythagorean proposition that the square on the longest side of a right angle is equal to the sum of the squares on the other two sides, are not projective but metrical, and are not handled by ordinary projective geometry. It was one of Cayley's greatest achievements in geometry to transcend the barrier which, before he leapt it, had separated projective from metrical pro- perties of figures. From his higher point of view metrical geo- metry also became projective, and the great power and flexi- bility of projective methods were shown to be applicable, by the introduction of 'imaginary' elements (for instance points whose co-ordinates involve V — 1) to metrical properties. Anyone who has done any analytic geometry will recall that two circles intersect in four points, two of which are always *imaginary'. (There are cases of apparent exception, for example concentric circles, but this is close enough for our purpose.) The fundamental notions in metrical geometry are the distance between two points and the angle between two lines. M.H.—VOL. n. K 433 *